Page 66 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 49
77. As long as we are speaking about the parts of some body or of some
material, we understand not ultimate or simple parts, of which indeed there
are none, but those that division really produces. Then, since by hypothesis
we admit material that is infinitely divisible, even a very small particle of
material can be dissected into many parts, but no number can be given that
is so large that a greater number of parts cut from that particle cannot be
exhibited. Hence the number of parts, indeed not ultimate parts, but those
that are still further divisible, that make up each body, is greater than
any number that can be given. Likewise, if the whole world is infinite,
the number of bodies making up the world is greater than any assignable
number. Since this is not a finite number, it follows that an infinite number
and a number greater than any assignable number are two ways of saying
the same thing.

78. Anyone who has gathered from this discussion any insight into the
infinite divisibility of matter will suffer none of the difficulties that people
commonly assign to this opinion. Nor will he be forced to admit anything
contrary to sound reasoning. On the other hand, anyone who denies that
matter is infinitely divisible will find himself in serious difficulties from
which he will in no way be able to extricate himself. These ultimate particles
are called by some atoms, by others monads or simple beings. The reason
why these ultimate particles admit no further division could be for two
possible reasons. The first is that they have no extension; the second is
that although they have extension, they are so hard and impenetrable that
no force is sufficient to dissect them. Whichever choice is made will lead to
equally difficult positions.

79. Suppose that ultimate particles lack any extension, so that they lack
any further parts: By this explanation the idea of simple beings is nicely
saved. However, it is impossible to conceive how a body can be constituted
by a finite number of particles of this sort. Suppose that a cubic foot of
matter is made up of a thousand simple beings of this kind, and that it is
actually cut up into one thousand pieces. If these pieces are equal, they will
each be one cubic finger; if they are not equal, some will be larger, some
smaller. One cubic finger will be a simple being, and we will be faced with
a great contradiction, unless by chance we want to say that there is one
simple being and the rest of the space is empty. In this way the continuity
of the body is denied, except that those philosophers completely banished
any vacuum from the world. If someone should object that the number
of simple beings contained in a cubic foot of matter is much more than
a thousand, absolutely nothing is gained. Any difficulty that follows from
the number one thousand will remain with any other number, no matter
how large. The inventor of the monad, a very acute man, LEIBNIZ, probed
this problem deeply, and finally decided that matter is infinitely divisible.
Hence, it is not possible to arrive at a monad before the body is actually